Đáp án:
Giải thích các bước giải:
a. \(\begin{array}{l}
BC = \sqrt {{3^2} + {3^2}} = 3\sqrt 2 \\
\overrightarrow {AB} .\overrightarrow {CM} = \overrightarrow {AB} .\overrightarrow {BC} = \left| {\overrightarrow {AB} } \right|.\left| {\overrightarrow {BC} } \right|.\cos \alpha = 3.3\sqrt 2 .\cos (180^\circ - 45^\circ )\\
= - 9
\end{array}\)
b. Có:
\(\begin{array}{l}
CN = \frac{2}{3}CB = 2\sqrt 2 \\
CM = BC = 3\sqrt 2 \\
\cos NCA = \cos 45^\circ = \frac{{\sqrt 2 }}{2} = \frac{{C{N^2} + A{C^2} - A{N^2}}}{{2.CN.AC}} = \frac{{{{(2\sqrt 2 )}^2} + {3^2} - A{N^2}}}{{2.2\sqrt 2 .3}}\\
\to AN = \sqrt 5 \\
\cos ACM = \cos 135^\circ = \frac{{C{M^2} + A{C^2} - A{M^2}}}{{2.CM.AC}} = \frac{{{{(3\sqrt 2 )}^2} + {3^2} - A{M^2}}}{{2.3\sqrt 2 .3}}\\
\to AM = 3\sqrt 5 \\
\to \cos NAC = \frac{{A{N^2} + A{C^2} - N{C^2}}}{{2.AN.AC}} = \frac{{5 + 9 - 8}}{{2.\sqrt 5 .3}} = \frac{{\sqrt 5 }}{5} \to \angle NAC \approx 63.435^\circ \\
\cos CAM = \frac{{A{M^2} + A{C^2} - M{C^2}}}{{2.AM.AC}} = \frac{{{{(3\sqrt 5 )}^2} + 9 - {{(3\sqrt 2 )}^2}}}{{2.3\sqrt 5 .3}} = \frac{{2\sqrt 5 }}{5} \to \angle CAM \approx 26.565^\circ \\
\to \angle NAM = \angle CAM + \angle NAC = 90^\circ \\
\to AN \bot AM
\end{array}\)
